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Properties

Label 2-336-7.2-c3-0-4
Degree $2$
Conductor $336$
Sign $-0.305 - 0.952i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 2.59i)3-s + (−0.119 + 0.207i)5-s + (−12.9 − 13.2i)7-s + (−4.5 + 7.79i)9-s + (4.04 + 7.01i)11-s + 43.7·13-s − 0.717·15-s + (30.4 + 52.6i)17-s + (−49.4 + 85.6i)19-s + (15.1 − 53.4i)21-s + (−106. + 185. i)23-s + (62.4 + 108. i)25-s − 27·27-s + 110.·29-s + (−40.0 − 69.2i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.0107 + 0.0185i)5-s + (−0.696 − 0.717i)7-s + (−0.166 + 0.288i)9-s + (0.110 + 0.192i)11-s + 0.933·13-s − 0.0123·15-s + (0.433 + 0.751i)17-s + (−0.597 + 1.03i)19-s + (0.157 − 0.555i)21-s + (−0.969 + 1.67i)23-s + (0.499 + 0.865i)25-s − 0.192·27-s + 0.706·29-s + (−0.231 − 0.401i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.305 - 0.952i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.439037350\)
\(L(\frac12)\) \(\approx\) \(1.439037350\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 - 2.59i)T \)
7 \( 1 + (12.9 + 13.2i)T \)
good5 \( 1 + (0.119 - 0.207i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-4.04 - 7.01i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 43.7T + 2.19e3T^{2} \)
17 \( 1 + (-30.4 - 52.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (49.4 - 85.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (106. - 185. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 110.T + 2.43e4T^{2} \)
31 \( 1 + (40.0 + 69.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-1.44 + 2.49i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 242.T + 6.89e4T^{2} \)
43 \( 1 + 367.T + 7.95e4T^{2} \)
47 \( 1 + (-44.6 + 77.4i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (5.70 + 9.88i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-200. - 346. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (240. - 416. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (120. + 208. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 978.T + 3.57e5T^{2} \)
73 \( 1 + (-311. - 538. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (272. - 472. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 845.T + 5.71e5T^{2} \)
89 \( 1 + (-101. + 176. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.19e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.24112677739310080515664896125, −10.29041411790489700947256204474, −9.753892718630573581312542085952, −8.605811215602421645435358662488, −7.71753844160104957964935398910, −6.52979278080337394503670768555, −5.53766098426031152498719381985, −3.98351915981060369519610405419, −3.42937756663863946666132343063, −1.52808843931193865042588283387, 0.50125577844024917112917132801, 2.28929889568573236395002451326, 3.34241501858837087071526852637, 4.84619434292691606971374937160, 6.26667424230859767125826747601, 6.72978179688841846292069279289, 8.277620290197451780438321102046, 8.753277359939360500822772599024, 9.845560248902488205665694671271, 10.87105427113544363562365052110

Graph of the $Z$-function along the critical line